Nuprl Lemma : member-concat
∀[T:Type]. ∀ll:T List List. ∀x:T. ((x ∈ concat(ll)) ⇐⇒ ∃l:T List. ((l ∈ ll) ∧ (x ∈ l)))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
concat: concat(ll),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
implies: P ⇒ Q,
concat: concat(ll),
top: Top,
iff: P ⇐⇒ Q,
false: False,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
or: P ∨ Q,
cand: A c∧ B,
guard: {T}
Lemmas referenced :
list_induction,
list_wf,
all_wf,
iff_wf,
l_member_wf,
concat_wf,
exists_wf,
reduce_nil_lemma,
false_wf,
nil_member,
nil_wf,
reduce_cons_lemma,
or_wf,
equal_wf,
cons_member,
cons_wf,
member_append,
append_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
productEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
productElimination,
addLevel,
allFunctionality,
impliesFunctionality,
because_Cache,
existsFunctionality,
andLevelFunctionality,
existsLevelFunctionality,
rename,
universeEquality,
unionElimination,
dependent_pairFormation,
inlFormation,
inrFormation,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}ll:T List List. \mforall{}x:T. ((x \mmember{} concat(ll)) \mLeftarrow{}{}\mRightarrow{} \mexists{}l:T List. ((l \mmember{} ll) \mwedge{} (x \mmember{} l)))
Date html generated:
2016_10_21-AM-10_33_57
Last ObjectModification:
2016_07_12-AM-05_46_02
Theory : list_1
Home
Index